To do this, we briefly describe how the algorithm of [1]} chooses initial functions \(f\) 's.
To make the description easier, we consider \(\pi \)  as an oriented face-edge path. Then let \(\pi ^{-1}\)  be the face-edge path
obtained from \(\pi \)  by reversing its orientation.
Let \(\phi \)  be the flow function of the given weak \(T\) -linkage of \(G^\textsf {p}\) .
For each violating path \(\pi \in \Pi \) , we let \(f_\pi \)  be the function that maps the first face to \(\phi (e)\) 
and maps all other faces to \(\epsilon \) , where \(e\)  be first edge in \(\pi \) .
Furthermore, we say two paths \(\pi \)  and \(\pi ^{\prime }\)  are co-viable if \(\bar{f}_{\pi } \vee \bar{f}_{\pi ^{\prime }}\)  is finite.
Recall that \(\bar{f}\)  is a smallest pre-feasible function of \(f\) .
